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Properties

Label 2-99-33.29-c1-0-0
Degree $2$
Conductor $99$
Sign $0.973 + 0.226i$
Analytic cond. $0.790518$
Root an. cond. $0.889111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.97 − 1.43i)2-s + (1.21 + 3.74i)4-s + (2.23 + 3.07i)5-s + (−0.349 + 0.113i)7-s + (1.45 − 4.48i)8-s − 9.24i·10-s + (2.97 − 1.46i)11-s + (−0.557 + 0.767i)13-s + (0.852 + 0.276i)14-s + (−2.92 + 2.12i)16-s + (−2.77 + 2.01i)17-s + (4.05 + 1.31i)19-s + (−8.78 + 12.0i)20-s + (−7.96 − 1.38i)22-s − 4.96i·23-s + ⋯
L(s)  = 1  + (−1.39 − 1.01i)2-s + (0.608 + 1.87i)4-s + (0.997 + 1.37i)5-s + (−0.132 + 0.0429i)7-s + (0.515 − 1.58i)8-s − 2.92i·10-s + (0.897 − 0.440i)11-s + (−0.154 + 0.212i)13-s + (0.227 + 0.0740i)14-s + (−0.731 + 0.531i)16-s + (−0.673 + 0.489i)17-s + (0.929 + 0.302i)19-s + (−1.96 + 2.70i)20-s + (−1.69 − 0.295i)22-s − 1.03i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(99\)    =    \(3^{2} \cdot 11\)
Sign: $0.973 + 0.226i$
Analytic conductor: \(0.790518\)
Root analytic conductor: \(0.889111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{99} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 99,\ (\ :1/2),\ 0.973 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.610847 - 0.0702166i\)
\(L(\frac12)\) \(\approx\) \(0.610847 - 0.0702166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-2.97 + 1.46i)T \)
good2 \( 1 + (1.97 + 1.43i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.23 - 3.07i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.349 - 0.113i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.557 - 0.767i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.77 - 2.01i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.05 - 1.31i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.96iT - 23T^{2} \)
29 \( 1 + (-0.767 - 2.36i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.84 + 2.06i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.21 + 6.83i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.840 + 2.58i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.88iT - 43T^{2} \)
47 \( 1 + (0.0195 + 0.00636i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.25 + 4.48i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (6.29 - 2.04i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.73 + 7.88i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.46T + 67T^{2} \)
71 \( 1 + (6.06 + 8.35i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.18 - 1.35i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.42 + 8.84i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.42 - 5.39i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 + (-12.2 - 8.86i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.86175202816943789627112334891, −12.43456421701771935432938056537, −11.24366822349596775124657809021, −10.60941801276476366426273049569, −9.673543847217256454933549325769, −8.848048393411771777133403997098, −7.30551575262942292852405807282, −6.20280971554546632574054600681, −3.35048469645583794992470785526, −2.01661134218978341531482207024, 1.37900142979280929225521552717, 4.98601444856056102550191966761, 6.14103586890522764363391448365, 7.35248933630106394609573582196, 8.684277806009354186720199807122, 9.374270673841449731739083399177, 9.995503481046031292483250331295, 11.69822611255280591270533312154, 13.07761710258488689702622939773, 14.08827860497354164192218021956

Graph of the $Z$-function along the critical line